一种基于PNP透视模型的合作目标位姿精度测量方法

摘要

本发明公开了一种基于PNP透视模型的合作目标位姿精度测量方法，包括如下步骤：在合作目标外表面安装N个视觉标记点，采用相机对视觉标记进行标记图像的拍摄；对合作目标与相机之间的位置姿态的初始值进行预先设定；其中第i个视觉标记点的三维空间坐标和有效焦距的水平、垂直分量，主点坐标的水平、垂直分量为未确定性系统误差，误差极限已知；第i个视觉标记点对应标记图像中视觉标记点中心二维坐标为随机误差，对应的误差极限已知；量化解析各项参数对位姿误差的影响权重系数；计算合作目标与相机之间的位置姿态的未确定性系统误差分量和随机误差分量并加权获得目标位姿测量误差，完成对目标位姿测量精度的预估。

主权项

一种基于PNP透视模型的合作目标位姿精度测量方法，其特征在于，步骤(1)、在所述合作目标外表面安装N个视觉标记点，采用相机对视觉标记进行标记图像的拍摄；步骤(2)、建立相机坐标系O_c‑X_cY_cZ_c：相机的光心为原点O_c，光轴为Z_c轴，相机所拍摄图像平面的水平和垂直方向分别为X_c轴和Y_c轴；建立关于合作目标的目标坐标系O_W‑X_WY_WZ_W：合作目标的质心为原点O_W，X_W、Y_W、Z_W与相机坐标系中的X、Y、Z轴相平行，且正方向均保持一致；其中第i个视觉标记点的三维空间坐标为(X_wi,Y_wi,Z_wi)及其对应标记图像中视觉标记点中心二维坐标为(u_i,v_i)；合作目标与相机之间的位置姿态为[t_x,t_y,t_z,α,β,γ]^T，则t_x,t_y,t_z分别为目标坐标系相对于相机坐标系中沿X_W、Y_W、Z_W轴的平移分量，α,β,γ分别为合作目标围绕目标坐标系中X_W、Y_W、Z_W轴的旋转角度；对于合作目标与相机之间的位置姿态的初始值进行预先设定；相机内参标定结果为：有效焦距的水平、垂直分量分别为f_x,f_y，主点坐标的水平、垂直分量分别为u₀,v₀；步骤(3)、将步骤(1)中的参数f_x,f_y,u₀,v₀以及X_wi,Y_wi,Z_wi确定为未确定性系统误差，f_x,f_y,u₀,v₀,X_wi,Y_wi,Z_wi各自对应的误差极限已知，且分别为e(f_x),e(f_y),e(u₀),e(v₀),e(X_wi),e(Y_wi),e(Z_wi)；将u_i,v_i确定为随机误差，对应的误差极限已知，且分别为δ(u_i),δ(v_i)；步骤(4)、量化解析各项参数对位姿误差的影响权重系数对于第i个视觉标记点，其位姿测量计算公式包括F_2i‑1和F_2i，二者分别为：$\left\{\begin{array}{c}{F}_{2i-1}(u_i,v_i,f_x,f_y,u_0,v_0,X_{\mathrm{wi}},Y_{\mathrm{wi}},Z_{\mathrm{wi}},\alpha ,\beta ,\gamma ,t_x,t_z)=(u_i-u_0)·(-\sin \beta ·X_{\mathrm{wi}}+\cos \beta \sin \alpha ·Y_{\mathrm{wi}}+\cos \beta \cos \alpha ·Z_{\mathrm{wi}}+t_z)\\ -f_x·[\cos \gamma ·\cos \beta ·X_{\mathrm{wi}}+(\cos \gamma ·\sin \beta ·\sin \alpha -\sin \gamma )·Y_{\mathrm{wi}}+(\cos \gamma ·\sin \beta ·\cos \alpha -\sin \gamma ·\sin \alpha )·Z_{\mathrm{wi}}+t_x];\\ F_{2i}(u_i,v_i,f_x,f_y,u_0,v_0,X_{\mathrm{wi}},Y_{\mathrm{wi}},Z_{\mathrm{wi}},\alpha ,\beta ,\gamma ,t_y,t_z)=(v_i-v_0)·(-\sin \beta ·X_{\mathrm{wi}}+\cos \beta \sin \alpha ·Y_{\mathrm{wi}}+\cos \beta \cos \alpha ·Z_{\mathrm{wi}}+t_z)\\ -f_y·[\sin \gamma ·\cos \beta ·X_{\mathrm{wi}}+(\sin \gamma ·\sin \beta ·\sin \alpha +\cos \gamma ·\cos \alpha )·Y_{\mathrm{wi}}+(\sin \gamma ·\sin \beta ·\cos \alpha +\cos \gamma ·\sin \alpha )·Z_{\mathrm{wi}}+t_y];\end{array}\right.$令：$M=\left\{\begin{array}{cccccc}\frac{{\partial F}_1}{\partial \alpha }& \frac{{\partial F}_1}{\partial \beta }& \frac{{\partial F}_1}{\partial \beta }& \frac{{\partial F}_1}{\partial t_x}& \frac{{\partial F}_1}{\partial t_y}& \frac{{\partial F}_1}{\partial t_z}\\ \frac{{\partial F}_2}{\partial \alpha }& \frac{{\partial F}_2}{\partial \beta }& \frac{{\partial F}_2}{\partial \gamma }& \frac{{\partial F}_2}{\partial t_x}& \frac{{\partial F}_2}{\partial t_y}& \frac{{\partial F}_2}{\partial t_z}\\ ...& ...& ...& ...& ...& ...\\ \frac{{\partial F}_{2i-1}}{\partial \alpha }& \frac{{\partial F}_{2i-1}}{\partial \beta }& \frac{{\partial F}_{2i-1}}{\partial \gamma }& \frac{{\partial F}_{2i-1}}{\partial t_x}& \frac{{\partial F}_{2i-1}}{\partial t_y}& \frac{{\partial F}_{2i-1}}{\partial t_z}\\ \frac{{\partial F}_{2i}}{\partial \alpha }& \frac{{\partial F}_{2i}}{\partial \beta }& \frac{{\partial F}_{2i}}{\partial \gamma }& \frac{{\partial F}_{2i}}{\partial t_x}& \frac{{\partial F}_{2i}}{\partial t_y}& \frac{{\partial F}_{2i}}{\partial t_z}\\ ...& ...& ...& ...& ...& ...\\ \frac{{\partial F}_{2N-1}}{\partial \alpha }& \frac{{\partial F}_{2N-1}}{\partial \beta }& \frac{{\partial F}_{2N-1}}{\partial \gamma }& \frac{{\partial F}_{2N-1}}{\partial t_x}& \frac{{\partial F}_{2N-1}}{\partial t_y}& \frac{{\partial F}_{2N-1}}{\partial t_z}\\ \frac{{\partial F}_{2N}}{\partial \alpha }& \frac{{\partial F}_{2N}}{\partial \beta }& \frac{{\partial F}_{2N}}{\partial \gamma }& \frac{{\partial F}_{2N}}{\partial t_x}& \frac{{\partial F}_{2N}}{\partial t_y}& \frac{{\partial F}_{2N}}{\partial t_z}\end{array}\right..$$X^{\prime }=·\left[\begin{array}{ccccccccccccc}\frac{\partial \alpha }{{\partial f}_x}& \frac{\partial \alpha }{{\partial f}_y}& \frac{\partial \alpha }{{\partial u}_0}& \frac{\partial \alpha }{{\partial v}_0}& \frac{\partial \alpha }{{\partial u}_i}& \frac{\partial \alpha }{{\partial v}_i}& \frac{\partial \alpha }{{\partial X}_{\mathrm{wi}}}& \frac{\partial \alpha }{{\partial Y}_{\mathrm{wi}}}& \frac{\partial \alpha }{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\\ \frac{\partial \beta }{{\partial f}_x}& \frac{\partial \beta }{{\partial f}_y}& \frac{\partial \beta }{{\partial u}_0}& \frac{\partial \beta }{{\partial v}_0}& \frac{\partial \beta }{{\partial u}_i}& \frac{\partial \beta }{{\partial v}_i}& \frac{\partial \beta }{{\partial X}_{\mathrm{wi}}}& \frac{\partial \beta }{{\partial Y}_{\mathrm{wi}}}& \frac{\partial \beta }{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\\ \frac{\partial \gamma }{{\partial f}_x}& \frac{\partial \gamma }{{\partial f}_y}& \frac{\partial \gamma }{{\partial u}_0}& \frac{\partial \gamma }{{\partial v}_0}& \frac{\partial \gamma }{{\partial u}_i}& \frac{\partial \gamma }{{\partial v}_i}& \frac{\partial \gamma }{{\partial X}_{\mathrm{wi}}}& \frac{\partial \gamma }{{\partial Y}_{\mathrm{wi}}}& \frac{\partial \gamma }{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\\ \frac{\partial t_x}{{\partial f}_x}& \frac{\partial t_x}{{\partial f}_y}& \frac{\partial t_x}{{\partial u}_0}& \frac{\partial t_x}{{\partial v}_0}& \frac{\partial t_x}{{\partial u}_i}& \frac{\partial t_x}{{\partial v}_i}& \frac{\partial t_x}{{\partial X}_{\mathrm{wi}}}& \frac{\partial t_x}{{\partial Y}_{\mathrm{wi}}}& \frac{\partial t_x}{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\\ \frac{\partial t_y}{{\partial f}_x}& \frac{\partial t_y}{{\partial f}_y}& \frac{\partial t_y}{{\partial u}_0}& \frac{\partial t_y}{{\partial v}_0}& \frac{\partial t_y}{{\partial u}_i}& \frac{\partial t_y}{{\partial v}_i}& \frac{\partial t_y}{{\partial X}_{\mathrm{wi}}}& \frac{\partial t_y}{{\partial Y}_{\mathrm{wi}}}& \frac{\partial t_y}{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\\ \frac{\partial t_z}{{\partial f}_x}& \frac{\partial t_z}{{\partial f}_y}& \frac{\partial t_z}{{\partial u}_0}& \frac{\partial t_z}{{\partial v}_0}& \frac{\partial t_z}{{\partial u}_i}& \frac{\partial t_z}{{\partial v}_i}& \frac{\partial t_z}{{\partial X}_{\mathrm{wi}}}& \frac{\partial t_z}{{\partial Y}_{\mathrm{wi}}}& \frac{\partial t_z}{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\end{array}\right]$$Y^{\prime }=\left[\begin{array}{ccccccccccccc}\frac{{\partial F}_1}{{\partial f}_x}& \frac{{\partial F}_1}{{\partial f}_y}& \frac{{\partial F}_1}{{\partial u}_0}& \frac{{\partial F}_1}{{\partial v}_0}& \frac{{\partial F}_1}{{\partial u}_i}& \frac{{\partial F}_1}{{\partial v}_i}& \frac{{\partial F}_1}{{\partial X}_{\mathrm{wi}}}& \frac{{\partial F}_1}{{\partial Y}_{\mathrm{wi}}}& \frac{{\partial F}_1}{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\\ \frac{{\partial F}_2}{{\partial f}_x}& \frac{{\partial F}_2}{{\partial f}_y}& \frac{{\partial F}_2}{{\partial u}_0}& \frac{{\partial F}_2}{{\partial v}_0}& \frac{{\partial F}_2}{{\partial v}_0}& \frac{{\partial F}_2}{{\partial v}_i}& \frac{{\partial F}_2}{{\partial X}_{\mathrm{wi}}}& \frac{{\partial F}_2}{{\partial Y}_{\mathrm{wi}}}& \frac{{\partial F}_2}{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\\ .& .& .& .& .& .& .& .& .& .& .& .& .\\ .& .& .& .& .& .& .& .& .& .& .& .& .\\ .& .& .& .& .& .& .& .& .& .& .& .& .\\ \frac{{\partial F}_{2i-1}}{{\partial f}_x}& \frac{{\partial F}_{2i-1}}{{\partial f}_y}& \frac{{\partial F}_{2i-1}}{{\partial u}_0}& \frac{{\partial F}_{2i-1}}{{\partial v}_0}& \frac{{\partial F}_{2i-1}}{{\partial v}_0}& \frac{{\partial F}_{2i-1}}{{\partial v}_i}& \frac{{\partial F}_{2i-1}}{{\partial X}_{\mathrm{wi}}}& \frac{{\partial F}_{2i-1}}{{\partial Y}_{\mathrm{wi}}}& \frac{{\partial F}_{2i-1}}{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\\ \frac{{\partial F}_{2i}}{{\partial f}_x}& \frac{{\partial F}_{2i}}{{\partial f}_y}& \frac{{\partial F}_{2i}}{{\partial u}_0}& \frac{{\partial F}_{2i}}{{\partial v}_0}& \frac{{\partial F}_{2i}}{{\partial v}_0}& \frac{{\partial F}_{2i}}{{\partial v}_i}& \frac{{\partial F}_{2i}}{{\partial X}_{\mathrm{wi}}}& \frac{{\partial F}_{2i}}{{\partial Y}_{\mathrm{wi}}}& \frac{{\partial F}_{2i}}{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\\ .& .& .& .& .& .& .& .& .& .& .& .& .\\ .& .& .& .& .& .& .& .& .& .& .& .& .\\ .& .& .& .& .& .& .& .& .& .& .& .& .\\ \frac{{\partial F}_{2N-1}}{{\partial f}_x}& \frac{{\partial F}_{2N-1}}{{\partial f}_y}& \frac{{\partial F}_{2N-1}}{{\partial u}_0}& \frac{{\partial F}_{2N-1}}{{\partial v}_0}& \frac{{\partial F}_{2N-1}}{{\partial v}_0}& \frac{{\partial F}_{2N-1}}{{\partial v}_i}& \frac{{\partial F}_{2N-1}}{{\partial X}_{\mathrm{wi}}}& \frac{{\partial F}_{2N-1}}{{\partial Y}_{\mathrm{wi}}}& \frac{{\partial F}_{2N-1}}{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\\ \frac{{\partial F}_{2N}}{{\partial f}_x}& \frac{{\partial F}_{2N}}{{\partial f}_y}& \frac{{\partial F}_{2N}}{{\partial u}_0}& \frac{{\partial F}_{2N}}{{\partial v}_0}& \frac{{\partial F}_{2N}}{{\partial v}_0}& \frac{{\partial F}_{2N}}{{\partial v}_i}& \frac{{\partial F}_{2N}}{{\partial X}_{\mathrm{wi}}}& \frac{{\partial F}_{2N}}{{\partial Y}_{\mathrm{wi}}}& \frac{{\partial F}_{2N}}{{\partial Z}_{\mathrm{wi}}}& ...& ...& ...& ...\end{array}\right],$其中X′为6行×2N列的矩阵，自第10列开始重复前9列内容；Y′为2N行×2N列的矩阵，且自第10列开始重复前9列内容；求解方程M·X′＝‑Y′中的X′，存在以下两种情况：①当2×N＝6时，利用最小二乘法求解，得到最优解为：X′＝(‑1)·M^‑1·Y′②当2×N＞6时，利用最小二乘法求解，得到最优解为：X′＝(‑1)·(M^T·M)^‑1·(M^T_·Y^′)步骤(5)、在所述随机误差u_i,v_i的影响下，合作目标与相机之间的位置姿态[t_x,t_y,t_z,α,β,γ]^T的随机误差分量分别为δ(t_x),δ(t_y),δ(t_z),δ(α),δ(β),δ(γ)；根据如下公式进行计算：$\left\{\begin{array}{c}\delta \left(\alpha \right)=\sqrt{\underset{i=1}{\overset{N}{\Sigma }}[{(\frac{\partial \alpha }{{\partial u}_i}·\delta \left(u_i\right))}^2+{(\frac{\partial \alpha }{{\partial v}_i}·\delta \left(v_i\right))}^2]}\\ \delta \left(\beta \right)=\sqrt{\underset{i=1}{\overset{N}{\Sigma }}[{(\frac{\partial \beta }{{\partial u}_i}·\delta \left(u_i\right))}^2+{(\frac{\partial \beta }{{\partial v}_i}·\delta \left(v_i\right))}^2]}\\ \delta \left(\gamma \right)=\sqrt{\underset{i=1}{\overset{N}{\Sigma }}[{(\frac{\partial \gamma }{{\partial u}_i}·\delta \left(u_i\right))}^2+{(\frac{\partial \gamma }{{\partial v}_i}·\delta \left(v_i\right))}^2]}\\ \delta \left(t_x\right)=\sqrt{\underset{i=1}{\overset{N}{\Sigma }}[{(\frac{\partial t_x}{{\partial u}_i}·\delta \left(u_i\right))}^2+{(\frac{\partial t_x}{{\partial v}_i}·\delta \left(v_i\right))}^2]}\\ \delta \left(t_y\right)=\sqrt{\underset{i=1}{\overset{N}{\Sigma }}[{(\frac{\partial t_y}{{\partial u}_i}·\delta \left(u_i\right))}^2+{(\frac{\partial t_y}{{\partial v}_i}·\delta \left(v_i\right))}^2]}\\ \delta \left(t_z\right)=\sqrt{\underset{i=1}{\overset{N}{\Sigma }}[{(\frac{\partial t_z}{{\partial u}_i}·\delta \left(u_i\right))}^2+{(\frac{\partial t_z}{{\partial v}_i}·\delta \left(v_i\right))}^2]}\end{array}\right.$在所述未确定性系统误差f_x,f_y,u₀,v₀,X_wi,Y_wi,Z_wi影响下，合作目标与相机之间的位置姿态[t_x,t_y,t_z,α,β,γ]^T的未确定性系统误差分量分别为e(t_x),e(t_y),e(t_z),e(α),e(β),e(γ)；根据如下公式进行计算：$\left\{\begin{array}{c}e\left(\alpha \right)=\sqrt{{[\frac{\partial \alpha }{{\partial f}_x}·e\left(f_x\right)]}^2+{[\frac{\partial \alpha }{{\partial f}_y}·e\left(f_y\right)]}^2+{[\frac{\partial \alpha }{{\partial u}_0}·e\left(u_0\right)]}^2+{[\frac{\partial \alpha }{{\partial v}_0}·e\left(v_0\right)]}^2+\underset{i=1}{\overset{N}{\Sigma }}\{{[\frac{\partial \alpha }{{\partial X}_{\mathrm{wi}}}·e\left(X_{\mathrm{wi}}\right)]}^2+{[\frac{\partial \alpha }{{\partial Y}_{\mathrm{wi}}}·e\left(Y_{\mathrm{wi}}\right)]}^2+{[\frac{\partial \alpha }{{\partial Z}_{\mathrm{wi}}}·e\left(Z_{\mathrm{wi}}\right)]}^2\}}\\ e\left(\beta \right)=\sqrt{{[\frac{\partial \beta }{{\partial f}_x}·e\left(f_x\right)]}^2+{[\frac{\partial \beta }{{\partial f}_y}·e\left(f_y\right)]}^2+{[\frac{\partial \beta }{{\partial u}_0}·e\left(u_0\right)]}^2+{[\frac{\partial \beta }{{\partial v}_0}·e\left(v_0\right)]}^2+\underset{i=1}{\overset{N}{\Sigma }}\{{[\frac{\partial \beta }{{\partial X}_{\mathrm{wi}}}·e\left(X_{\mathrm{wi}}\right)]}^2+{[\frac{\partial \beta }{{\partial Y}_{\mathrm{wi}}}·e\left(Y_{\mathrm{wi}}\right)]}^2+{[\frac{\partial \beta }{{\partial Z}_{\mathrm{wi}}}·e\left(Z_{\mathrm{wi}}\right)]}^2\}}\\ e\left(\gamma \right)=\sqrt{{[\frac{\partial \gamma }{{\partial f}_x}·e\left(f_x\right)]}^2+{[\frac{\partial \gamma }{{\partial f}_y}·e\left(f_y\right)]}^2+{[\frac{\partial \gamma }{{\partial u}_0}·e\left(u_0\right)]}^2+{[\frac{\partial \gamma }{{\partial v}_0}·e\left(v_0\right)]}^2+\underset{i=1}{\overset{N}{\Sigma }}\{{[\frac{\partial \gamma }{{\partial X}_{\mathrm{wi}}}·e\left(X_{\mathrm{wi}}\right)]}^2+{[\frac{\partial \gamma }{{\partial Y}_{\mathrm{wi}}}·e\left(Y_{\mathrm{wi}}\right)]}^2+{[\frac{\partial \gamma }{{\partial Z}_{\mathrm{wi}}}·e\left(Z_{\mathrm{wi}}\right)]}^2\}}\\ e\left(t_x\right)=\sqrt{{[\frac{\partial t_x}{{\partial f}_x}·e\left(f_x\right)]}^2+{[\frac{\partial t_x}{{\partial f}_y}·e\left(f_y\right)]}^2+{[\frac{\partial t_x}{{\partial u}_0}·e\left(u_0\right)]}^2+{[\frac{\partial t_x}{{\partial v}_0}·e\left(v_0\right)]}^2+\underset{i=1}{\overset{N}{\Sigma }}\{{[\frac{\partial t_x}{{\partial X}_{\mathrm{wi}}}·e\left(X_{\mathrm{wi}}\right)]}^2+{[\frac{\partial t_x}{{\partial Y}_{\mathrm{wi}}}·e\left(Y_{\mathrm{wi}}\right)]}^2+{[\frac{\partial t_x}{{\partial Z}_{\mathrm{wi}}}·e\left(Z_{\mathrm{wi}}\right)]}^2\}}\\ e\left(t_y\right)=\sqrt{{[\frac{\partial t_y}{{\partial f}_x}·e\left(f_x\right)]}^2+{[\frac{\partial t_y}{{\partial f}_y}·e\left(f_y\right)]}^2+{[\frac{\partial t_y}{{\partial u}_0}·e\left(u_0\right)]}^2+{[\frac{\partial t_y}{{\partial v}_0}·e\left(v_0\right)]}^2+\underset{i=1}{\overset{N}{\Sigma }}\{{[\frac{\partial t_y}{{\partial X}_{\mathrm{wi}}}·e\left(X_{\mathrm{wi}}\right)]}^2+{[\frac{\partial t_y}{{\partial Y}_{\mathrm{wi}}}·e\left(Y_{\mathrm{wi}}\right)]}^2+{[\frac{\partial t_y}{{\partial Z}_{\mathrm{wi}}}·e\left(Z_{\mathrm{wi}}\right)]}^2\}}\\ e\left(t_z\right)=\sqrt{{[\frac{\partial t_z}{{\partial f}_x}·e\left(f_x\right)]}^2+{[\frac{\partial t_z}{{\partial f}_y}·e\left(f_y\right)]}^2+{[\frac{\partial t_z}{{\partial u}_0}·e\left(u_0\right)]}^2+{[\frac{\partial t_z}{{\partial v}_0}·e\left(v_0\right)]}^2+\underset{i=1}{\overset{N}{\Sigma }}\{{[\frac{\partial t_z}{{\partial X}_{\mathrm{wi}}}·e\left(X_{\mathrm{wi}}\right)]}^2+{[\frac{\partial t_z}{{\partial Y}_{\mathrm{wi}}}·e\left(Y_{\mathrm{wi}}\right)]}^2+{[\frac{\partial t_z}{{\partial Z}_{\mathrm{wi}}}·e\left(Z_{\mathrm{wi}}\right)]}^2\}}\end{array}\right.$步骤(6)采用目标位姿随机误差分量与目标位姿未确定性系统误差分量依据如下公式相加获得目标位姿测量误差，完成对目标位姿测量精度的预估：$\left\{\begin{array}{c}{\left(\mathrm{\Delta \alpha }\right)}_{\mathrm{total}}^2={\left[e\left(\alpha \right)\right]}^2+\frac{1}{n}·{\left[\delta \left(\alpha \right)\right]}^2;\\ {\left(\mathrm{\Delta \beta }\right)}_{\mathrm{total}}^2={\left[e\left(\beta \right)\right]}^2+\frac{1}{n}·{\left[\delta \left(\beta \right)\right]}^2;\\ {\left(\mathrm{\Delta \gamma }\right)}_{\mathrm{total}}^2={\left[e\left(\gamma \right)\right]}^2+\frac{1}{n}·{\left[\delta \left(\gamma \right)\right]}^2;\\ {\left(\Delta t_x\right)}_{\mathrm{total}}^2={\left[e\left(t_x\right)\right]}^2+\frac{1}{n}·{\left[\delta \left(t_x\right)\right]}^2;\\ {\left(\Delta t_y\right)}_{\mathrm{total}}^2={\left[e\left(t_y\right)\right]}^2+\frac{1}{n}·{\left[\delta \left(t_y\right)\right]}^2;\\ {\left(\Delta t_z\right)}_{\mathrm{total}}^2={\left[e\left(t_z\right)\right]}^2+\frac{1}{n}·{\left[\delta \left(t_z\right)\right]}^2;\end{array}\right.,$其中n任意取值。